The polarization dependence of γγ absorption - implications for γ-ray bursts and blazars

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The Astrophysical Journal, 795:35 (4pp), 2014 November 1 doi:10.1088/0004-637X/795/1/35 C

THE POLARIZATION DEPENDENCE OF γ γ ABSORPTION—IMPLICATIONS

FOR γ -RAY BURSTS AND BLAZARS

M. B ¨ottcher

Centre for Space Research, North-West University, Potchefstroom 2520, South Africa;Markus.Bottcher@nwu.ac.za

Received 2014 July 25; accepted 2014 September 4; published 2014 October 9

ABSTRACT

This paper presents an analysis of the dependence of the opacity for high-energy γ -rays to γ γ absorption by low-energy photons on the polarization of the γ -ray and target photons. This process has so far only been considered using the polarization-averaged γ γ absorption cross section. It is demonstrated that in the case of polarized

γ-ray emission, subject to source-intrinsic γ γ absorption by polarized target photons, this may lead to a slight

overestimation of the γ γ opacity by up to∼10% in the case of a perfectly ordered magnetic field. Thus, for realistic astrophysical scenarios with partially ordered magnetic fields, the use of the polarization-averaged γ γ cross section is justified for practical purposes, such as estimates of minimum Doppler factors inferred for γ -ray bursts and blazars, based on γ γ transparency arguments; this paper quantifies the small error incurred by the unpolarized-radiation approximation. Furthermore, it is shown that polarization-dependent γ γ absorption of initially polarized

γ-rays can lead to a slight increase in the polarization beyond the spectral break caused by γ γ absorption. This

amount is distinctly different from the change in polarization expected if the same spectral break were produced by a break in the underlying electron distribution. This may serve as a diagnostic of whether γ γ absorption is relevant in sources such as γ -ray bursts and blazars where the γ -ray emission may be intrinsically highly polarized. Key words: galaxies: jets – gamma-ray burst: general – gamma rays: galaxies – radiation mechanisms: non-thermal – relativistic processes

Online-only material: color figures

1. INTRODUCTION

It has long been recognized (e.g., Gould & Schr´eder1967) that high-energy γ -rays from astronomical sources are subject to γ γ absorption by low-energy target photons if the energy

threshold for pair production γt(1 − μ) 2 is fulfilled.

Here γ and t are the energies of the γ -ray and the soft

tar-get photon, respectively, normalized to the electron rest-mass

energy, = E/(mec2), and μ = cos θ is the cosine of the

collision angle. This process is expected to limit the very-high-energy (VHE) γ -ray horizon out to which VHE γ -rays may be detectable by ground-based γ -ray observatories due to γ γ absorption in the extragalactic background light (EBL;

e.g., Stecker et al. 1992; Finke et al. 2010, and

refer-ences therein). The apparent absence of γ γ absorption signa-tures also provides evidence for relativistic beaming in γ -ray

bursts (GRBs; e.g., Baring 1993) and blazars (e.g., Dondi &

Ghisellini1995).

When considering the effects of γ γ absorption, to the au-thor’s knowledge, all previous works have used the polarization-averaged cross section for γ γ absorption (or simplified approx-imations derived from it),

σ_{γ γ}ave= 3 16σT(1− β 2₎ × (3− β4) ln 1 + β 1− β − 2 β (2 − β2₎ (1) (Jauch & Rohrlich1976), where

β =

1− 2

γt(1− μ)

(2) is the normalized (to the speed of light) velocity of the electron and positron produced in the center-of-momentum frame of the

collision. In the case of γ γ absorption of cosmological γ -rays by the EBL, the use of the polarization-averaged cross section is reasonable as the EBL is expected to be, on average, unpolarized on cosmological scales.

However, this may not be the case when considering source-intrinsic γ γ absorption in GRBs and blazars. In GRBs, the X-ray through γ -ray emission is commonly interpreted as syn-chrotron radiation by shock-accelerated electrons, possibly with an admixture of thermal radiation from a photosphere of the initial fireball. In an ordered magnetic field, synchrotron radi-ation is expected to be polarized, and Compton scattering in the photosphere of structured GRB outflows may also result in non-zero polarization of a possible photospheric emission

com-ponent (Lundman et al.2014). In blazars, the low-energy

emis-sion, potentially acting as targets for intrinsic γ γ absorption, is generally agreed to be produced by synchrotron radiation of relativistic electrons (see, e.g., B¨ottcher 2007for a review of blazar emission models). In the case of synchrotron self-Compton (SSC) radiation or hadronic emission scenarios for the γ -ray emission, the γ -rays are also expected to be polarized (Zhang & B¨ottcher2013). Therefore, when considering the in-trinsic γ γ opacity of high-energy γ -ray sources, the effects of polarization may not be negligible.

The study of potential effects of the polarization dependence of the γ γ opacity in GRBs and blazars is the aim of this

pa-per. Section 2 contains a brief discussion of the

polarization-dependent cross section for γ γ absorption. Section3presents

a simple model scenario to illustrate the potential impact of the polarization dependence of the γ γ absorption cross section in the case of intrinsic absorption of polarized γ -rays by a tar-get photon field with the same polarization geometry as the

γ-rays; this is generally expected when γ -rays and target

pho-tons are produced co-spatially. Section4contains a discussion of the results.

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The Astrophysical Journal, 795:35 (4pp), 2014 November 1 B ¨ottcher 1 2 3 4 5 6 7 8 9 10 ε₁ ε₂ (1 − μ) / 2 0.0 0.5 1.0 1.5 2.0 σγγ [10 -25 cm 2 ] Average Parallel Perpendicular

Figure 1. Polarization-dependent γ γ absorption cross section as a function of

center-of-momentum electron energy squared, γ2

cm= γt(1− μ)/2, for the

case of parallel and perpendicular polarization directions of the γ -ray and target photons, respectively.

(A color version of this figure is available in the online journal.)

2. THE POLARIZATION-DEPENDENT

γ γ ABSORPTION CROSS SECTION

The polarization-dependent cross section for γ γ absorption

has been calculated by Breit & Wheeler (1934). Using the

same nomenclature as employed in (Jauch & Rohrlich1976;

e.g., Equations (1) and (2)), the cross section σ_{γ γ} for the case of parallel linear-polarization vectors of the γ -ray and target photon is given by σ_{γ γ} = 3 16σT(1− β 2₎ × 5 + 2 β2_{− 3 β}4 2 ln 1 + β 1− β − β (5 − 3 β2 ) , (3) and σ_{γ γ}⊥ for the case of perpendicular polarization vectors:

σ_{γ γ}⊥ = 3 16σT(1− β 2₎ × 7− 2 β2_{− β}4 2 ln 1 + β 1− β − β (3 − β2₎ . (4) One can easily verify that (σγ γ + σγ γ⊥)/2 = σγ γave according

to Equation (1). These cross sections as a function of

center-of-momentum energy are plotted in Figure1. It can be seen

that the cross section for the case of perpendicular polarization directions (1) peaks at lower energies than the average and the parallel cases and (2) peaks at a value about 1.5 times higher the peak cross section for the parallel case.

This means that the γ γ absorption of polarized γ -ray photons by target photons with identical polarization direction is sup-pressed compared to the unpolarized case (and compared to the case of absorption on target photons with polarization direction perpendicular to that of the γ -ray photon). This may be impor-tant in cases where polarized γ -rays are produced co-spatially with synchrotron target photons, with preferred electric-field vector orientations perpendicular to a globally ordered mag-netic field in the emission region. If the γ -rays are co-spatially produced, e.g., by SSC scattering, proton synchrotron radiation,

or synchrotron emission of secondary particles in photo-pion-induced cascade processes, then they are expected to have the same polarization direction as the target electron–synchrotron photons (Zhang & B¨ottcher2013). In this case, we expect two consequences of the polarization-dependence of the cross sec-tions (Secsec-tions3and4). (1) The overall γ γ opacity is reduced compared to γ γ absorption by unpolarized photons. (2) The

degree of polarization Π is expected to increase in a partially

self-absorbed regime (see, e.g., Panaitescu et al. 2014 for a

discussion of possible signatures of partially self-absorbed SSC emission in GRBs), because the dominant polarization direction of the γ -ray beam will be less affected by γ γ absorption since it is primarily interacting with the sub-dominant polarization direction of the target photon field and vice versa. These effects will be illustrated and studied more quantitatively with a sim-ple toy model of a synchrotron-dominated γ -ray source in the next section.

3. EFFECTS ON A SYNCHROTRON-DOMINATED

γ-RAY SOURCE

In this section, we investigate the potential effects of polarization-dependent γ γ absorption in an idealized case of a synchrotron-dominated γ -ray source, as commonly assumed for the non-thermal emission from GRBs. The model assumes a spherical emission region containing a perfectly ordered mag-netic field and considers a non-thermal electron synchrotron spectrum characterized by an energy index α, i.e., an

emis-sion coefficient jν ∝ ν−α, extending from 1 = 10−6 without

cut-off into the γ -ray regime. Additional polarized high-energy and VHE γ -ray photons may be produced, e.g., by proton-synchrotron radiation or proton-synchrotron emission of secondary leptons produced in photo-pion pair cascades (e.g., Zhang &

B¨ottcher2013). The synchrotron emissivity is normalized to a

total synchrotron compactness = σTLsy/(4π Rmec3) ∼

1, so that γ γ absorption effects are expected to become impor-tant at high energies. Based on the spectral index α, the syn-chrotron spectrum is characterized by a degree of polarization,

Π ≡ j⊥− j

j_⊥+ j = α+ 1

α+ 5/3, (5)

from which we find the emission coefficients with electric-field vector orientations perpendicular and parallel to the magnetic field, j_⊥ = j (1 + Π)/2 and j= j (1 − Π)/2, respectively. For the case study in this section, let us consider the γ γ opacity of a synchrotron γ -ray beam emitted perpendicular to the B field. Due to the dependence of the synchrotron emissivity on the pitch angle χ of relativistic particles, jν(χ ) ∝ sin2χ,

the majority of polarized synchrotron photons is expected to

be emitted in this direction (χ = π/2). Let us further choose

the direction of propagation of the γ -ray to be the z-axis, and the B field to be along the y-axis. We define θ as the angle that a target photon’s momentum makes with the z-axis (and, thus, the γ -ray photon momentum), and φ as the azimuthal angle

around the z-axis, with φ = 0 in the (y, z) plane. The γ -ray

photon then has polarization vectors given by ˆsγ ,⊥ = ˆx and

ˆsγ ,= ˆy. One then finds that the polarization vector st,parallel

to the magnetic field projection onto the plane perpendicular to the direction of propagation of any target photon interacting with the γ -ray from an angle (θ , φ) forms an angle ψ with sγ ,

given by cos ψ = ˆsγ ,· ˆst, =

1− sin2_θ _cos2_φ_{. Using this} result, one can find the γ γ absorption coefficients for γ -rays, with electric-field vectors parallel and perpendicular to the 2

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The Astrophysical Journal, 795:35 (4pp), 2014 November 1 B ¨ottcher 100 101 E_γ / m_ec2 1 τγγ Parallel Perpendicular Average

Figure 2. Polarization-dependent γ γ opacity as a function of γ -ray photon

en-ergy, in a synchrotron-dominated source, for photons propagating perpendicular to the magnetic field, in a source with compactness = 1, with synchrotron spectral index α= 0.5. The figure illustrates that the opacity for γ -rays with electric-field vectors perpendicular to the B field (i.e., the dominant polariza-tion direcpolariza-tion) is about 10% smaller than for photons with E field vectors parallel to B.

magnetic field, as κ_ν⊥= _∞ 0 d 4π dΩ (1 − μ) · σ_{γ γ} (n⊥[,Ω] cos2ψ+ n[,Ω] sin2ψ) + σγ γ⊥(n[,Ω] cos2ψ+ n⊥[,Ω] sin2ψ) (6) and κ_ν= _∞ 0 d 4π dΩ (1 − μ) · σ_{γ γ} (n[,Ω] cos2ψ+ n⊥[,Ω] sin2ψ) + σγ γ⊥(n⊥[,Ω] cos2ψ+ n[,Ω] sin2ψ) , (7)

where the photon densities n⊥ and n are evaluated based on

the emissivities jand j_⊥as outlined above.

Figure2shows the resulting γ γ opacities for a source with

compactness = 1 and a synchrotron spectral index of α = 0.5.

Both the polarization-dependent and the polarization-averaged

opacities show the well-known energy dependence τγ γ ∝ α. As

expected, the γ γ opacity for photons with parallel polarization direction is larger than that for photons with perpendicular (the dominant) polarization direction. However, the effect is smaller than the difference in the peak values of the respective cross sections, since the target photon field is not 100% polarized, thus mitigating the effect. Still, the difference between the opacities in the two polarization directions is about 10% in this case.

Given an expected optically thin synchrotron flux Fint

ν

emerg-ing from the spherical synchrotron source considered here, the

emerging spectrum Fobs

ν including the effects of γ γ absorption

is calculated separately for both polarization directions as F_νobs= F_νint1− e

−τγ γ

τγ γ

. (8)

The degree of polarization Π of the emerging spectrum is

then evaluated based on the emerging fluxes with parallel and

10-3 10-2 10-1 100 101 102 103 104 105 106 E_γ / m_ec2 1e+41 1e+42 1e+43 ν Fν

[arbitrary units] Parallel

Perpendicular Total 0.7

0.72

Π [%]

Figure 3. Polarization-dependent flux spectrum (lower panel) and degree of

polarization as a function of photon energy (upper panel). Parameters are the same as for Figure2.

2 4 6 8 10 12 (τave. - τperp ) / τave. [%] -0.5 0 0.5 1 1.5 Spectral index _α -4 -2 0 2 4 6 ΔΠ [%] γγ-absorption e-spectrum break

Figure 4. Upper panel: ratio of unpolarized (i.e., average) to polarized

(perpendicular) γ γ opacity as a function of spectral index α. Lower panel: change of the degree of polarizationΠ across the spectral break, as a function of spectral index α, expected for a break caused by polarization-dependent γ γ absorption (circles) and for a break caused by a break in the underlying electron spectrum (squares).

perpendicular polarization directions, i.e.,Π = (F_⊥−F)/(F_⊥+ F). The result for our baseline example is plotted in Figure3.

One sees the expected spectral breakΔαγ = αtarget = α = 0.5

in accordance with Equation (8). The top panel illustrates the

effect mentioned in the previous section, that the dominant (perpendicular) polarization direction is less affected by γ γ absorption than the sub-dominant (parallel) one, leading to an

increase of the degree of polarization Π toward the optically

thick regime.

While the change of the degree of polarization,ΔΠ, expected

from polarization-dependent γ γ absorption is a rather small effect (ΔΠ = 2.7%), it can be compared to the change expected if the associated spectral break is due to a break in the underlying

electron distribution. In that case, one expects a changeΔΠ =

Δ([α + 1]/[α + 5/3]). For a break from α = 0.5 → 1, this would

yield a change ofΔΠ = 5.8%.

Figure4illustrates how these results depend on the spectral

index α of the synchrotron spectrum. The upper panel shows 3

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The Astrophysical Journal, 795:35 (4pp), 2014 November 1 B ¨ottcher

that the use of the average (unpolarized) γ γ opacity may overestimate the actual opacity for polarized γ -rays by up to ∼10% for a relatively steep intrinsic spectrum. The lower panel

compares the change in γ -ray polarizationΠ across the spectral

break between the two cases—one caused by γ γ absorption and the other by a break in the underlying electron distribution. While the polarization change due to γ γ absorption is always

expected to be at the 4% level, a break in the underlying

electron distribution may cause much larger changes. Below we will discuss whether this may be used as a diagnostic of the importance of γ γ absorption in the formation of the high-energy spectra of GRBs and blazars.

4. SUMMARY AND DISCUSSION

The main results of the study presented in the previous sections can be summarized as follows.

1. The use of unpolarized (average) γ γ opacity may

overesti-mate the actual γ γ opacity by a small amount, up to∼10%

in cases where high-energy γ -rays and target photons have identical preferred polarization directions, and the magnetic field is perfectly ordered. This effect becomes larger with the increasing spectral index of the target photon field. 2. Polarization-dependent γ γ absorption leads to a spectral

break in the emerging γ -ray spectrum, which is

accompa-nied by a small increase of the percentage polarizationΠ,

which is, for spectral indices α 0.2, smaller than the

expected change in polarization, resulting from a break in the underlying electron distribution.

Both in the case of GRBs and blazars, the target photon field for γ γ absorption is most likely of synchrotron origin, and therefore expected to be polarized. The non-thermal γ -ray emission from GRBs is also commonly attributed to synchrotron emission (and possibly SSC radiation), while that of blazars may be due to SSC emission or proton-induced processes, such as proton-synchrotron or synchrotron radiation from photo-pion-induced secondaries. In those cases, the γ -ray emission from blazars is also expected to be polarized. The study presented here has shown that the use of the unpolarized γ γ opacity may slightly overestimate, e.g., the minimum Lorentz factors of blazars and GRBs.

In our simple toy model, a perfectly ordered magnetic field has been assumed. From the observed optical polarization, e.g.,

of blazars, reaching maximum values of Π 40%, one can

infer that due to a partial disorder in the B field, the degree of polarization of the γ -ray and target photon fields is reduced to at most about 50% of the level expected for a perfectly ordered B field. Therefore, realistically, one may expect that γ γ opacities

may be overestimated by no more than∼5%, which—for most

practical purposes—is a sufficiently small error to justify the use of the polarization-averaged γ γ cross section.

An additional, simplifying assumption made in our toy model was the extension of the polarized target photon field into the γ -ray regime without any cut-off. A high-energy cut-off of the synchrotron target photon spectrum would result in an increased degree of polarization at and beyond the (normalized) cut-off energy cut. This will result in a larger effect of the polarization dependence of the γ γ absorption cross section at

γ-ray photon energies γ 1/cut, where the overestimation

of the γ γ opacity when using the polarization-averaged cross section would become more severe than discussed above.

The measurement of high-energy polarization is a very challenging task. However, satellite-borne instruments, such as SPectrometer on IMAGER and Imager on Board the INTEGRAL Satellite, have already been used successfully to constrain the hard X-ray/soft γ -ray polarization from GRBs

(Dean et al. 2008; Forot et al. 2008). Design studies for

the upcoming ASTRO-H mission suggest that it may also be able to detect polarization in the 50–200 keV energy band

(Tajima et al. 2010). It has also been suggested that the

Large Area Telescope (LAT) on board the Fermi Gamma-Ray Space Telescope may be able to detect γ -ray

polariza-tion in the energy range∼30–200 MeV when considering

pair-conversion events occurring in the silicon layers of the detec-tor by taking advantage of the polarization-dependent direc-tion of modirec-tion of the electron–positron pairs produced in the γ-ray pair-conversion process (B¨uhler et al.2010). For bright

γ-ray sources, degrees of polarization down to∼10% may be

detectable. However, the feasibility of such γ -ray polarization measurements with Fermi-LAT is highly controversial. The

pro-posed PANGU and GAMMA-LIGHT mission (Wu et al.2014;

Morselli et al.2014), due to the absence of any tungsten

con-version layers in its design, may provide substantial progress in the ability to measure γ -ray polarization. With such advances, it may be feasible to detect the γ -ray polarization of GRBs and blazars. This will open up the avenue to (1) more precisely de-termine the expected γ γ opacity constraints relevant to these sources and (2) identify the nature of spectral breaks in the γ -ray spectra of GRBs and blazars, which will afford deeper insight into the nature of the underlying electron distribution and, hence, the mechanisms leading to the acceleration of particles to ultra-relativistic energies in the ultra-relativistic jets of GRBs and blazars. The author thanks Haocheng Zhang for stimulating discus-sions and the anonymous referee for a helpful and constructive report which helped improve the manuscript. He acknowledges support from the South African Department of Science and Technology through the National Research Foundation under NRF SARChI Chair grant No. 64789.

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